Scaling up multi-mode entanglement generated by mode swapping

arXiv:2609.15040 · quant-ph · Submitted 2026-09-14 · Read on arXiv

Listen

Transcript

Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: I'm Kai.

Mira: And I'm Mira, here with Lev.

Kai: We're looking at the paper: "Scaling up multi-mode entanglement generated by mode swapping".

Mira: The gist Entanglement between two local multi-photon multi-mode systems can be generated by swapping a pair of modes between the two local multi-mode systems.

Kai: First, who's behind it and why it matters.

Title and authors: Kai: The paper's title is "Scaling up multi-mode entanglement generated by mode swapping," and it's about showing how you can scale this up to three or more modes. The authors are Yuki Kodama and Holger F. Hofmann from the Graduate School of Advanced Science and Engineering at Hiroshima University, Japan.

Mira: So, the core idea is that entanglement between two local multi-photon multi-mode systems can be generated by swapping a pair of modes between those systems. That's what they put in their abstract; it’s a very direct statement about the mechanism.

Lev: The authors are focusing on efficiency here, because scaling up these operations usually means dealing with much more noise and complexity in the actual physical implementation on a quantum computer.

Kai: Right, because they aren't just looking at two modes anymore; they're considering systems with three or more modes, which is what makes the experiment much harder to set up and measure.

Mira: They suggest that the choice of photon number inputs in those local systems introduces a new degree of freedom into the non-local interference effects that show up in the output photon number statistics.

Lev: That implies that if you're trying to control or predict these output statistics, you need to account for how those initial input numbers influence the final result, not just assume a fixed state.

The paper's summary: Kai: The paper summarizes their setup by describing two systems, A and B, each having three local modes. They apply a local unitary transformation before the swap and then the inverse transformation after to make sure that when they exchange those modes, they are equal superpositions of each other.

Mira: They use a discrete Fourier transform to ensure that these swapped modes are indeed equal superpositions of the input modes from both local systems, which is key for making sure you get those specific non-local correlations.

Lev: But the analysis gets really interesting when they look at a specific asymmetric three-photon input state, in = two hundred A one hundred B, and then they calculate the final output state after applying the swap unitary operation <ref:2609.15040#pg2,specific asymmetric three-photon input state>.

Kai: After getting that final state, MS, they look at the photon number distribution and post-select to see what happens when you only keep terms without photon transfer or terms describing a reciprocal transfer of one photon in each direction.

Mira: They found that the success probability for this specific post-selection is P twenty-one = eight/twenty-seven + four/twenty-seven which equals four/nine. That tells us the odds of getting that specific type of correlated output.

Lev: The analysis shows that when you look at the local photon number distribution for this input, it’s (N A, N B) = (two one), and they show that this is preserved either when all three photons are in components orthogonal to the swapped modes or when one photon is exchanged in each direction between systems A and B <ref:2609.15040#pg1>.

The paper's improvements: Kai: One of the main improvements they point out is that the mode swapping operation actually only affects the two-mode subspace spanned by those two swapped modes, which means its effect appears through the difference between those swapped-mode operators, B - A.

Mira: That's a big piece of information because it allows them to design quantum circuits where the entanglement generation is explicitly mapped onto a controlled, localized operation happening only between specific output modes.

Lev: For running this on real hardware, that localization is crucial; if we can isolate the control to just those two modes, we reduce the amount of ancillary operations needed for the swap itself.

Kai: They also show that when you look at the joint photon number probabilities P(N A, N B) for that input state two hundred A one hundred B, the conditional probability of getting the same output distribution as your input is four/nine considering only results where N A = two and N B = one.

Mira: They also noted that the zero probabilities for states like two hundred A ten B and two hundred A one hundred one B indicate complete destructive interference between the no-swap and swap amplitudes.

Lev: That interference structure is what suggests the potential for entanglement generation by mode swapping, which gives us a blueprint for how we might want to design future quantum circuits with more modes and photons.

Conclusion: Kai: So, to wrap up the paper "Scaling up multi-mode entanglement generated by mode swapping," it shows that you can indeed generate entanglement between local multi-photon systems by swapping modes, and the input photon numbers give you a new way to control those non-local interference effects.

Mira: The key finding is that the post-selected state twenty-one exhibits strong correlations in the output photon number distributions between the input modes of your two local systems when you use this method.

Lev: For us on the hardware side, it means we need to be very careful about our measurement protocols because you have to post-select correctly at that four/nine probability rate to actually see those specific correlations.

Kai: It gives us a clear path forward for extending this analysis into optical quantum circuits with even larger numbers of modes and photons, showing how the DFT helps connect each local output mode to modes in the other system.

Mira: The paper provides a theoretical framework that shows exactly how these systems can be entangled through mode swapping, which is really useful context for understanding multiphoton interference in linear optics.

Lev: I just want to say that while this is a great theoretical basis, the next big step will be figuring out the exact gate sequence needed to implement this swap reliably on actual superconducting qubits or photons.

Kai: It definitely points toward building those larger circuits we've been trying to design for quantum communication and metrology applications.

Graduate School of Advanced Science and Engineering, Hiroshima University

quant-ph

Submitted: 2026-09-14

Updated: 2026-09-14

Comments: 6 pages, 1 figure. Presented at SPIE Optics + Photonics 2026

Journal ref: Proc. SPIE 14228, Quantum Communications and Quantum Imaging XXIV, 1422806 (2026)

DOI: 10.1117/12.3114886

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 64/100

The gist: The gist Entanglement between two local multi-photon multi-mode systems can be generated by swapping a pair of modes between the two local multi-mode systems.

Key concepts

Mode Swapping: This operation involves exchanging one spatial mode from each of two separate multi-mode systems. It is an extension of the Hong-Ou-Mandel effect to spatially separated systems, introducing non-local correlations between the two original systems by linking their modes through this exchange.
Asymmetric Three-Photon Input State: The experiment uses a specific input state where System A has two photons in mode 0 and zero elsewhere (|200>A), and System B has one photon in mode 0 and zero elsewhere (|100>B). This specific configuration is chosen to reveal the non-local interference effects when the modes are swapped.
Discrete Fourier Transform (DFT): The DFT is used to define the local unitary transformations. It ensures that the modes being exchanged between systems A and B are equal superpositions of their respective input modes. This mathematical tool helps establish a clear connection between the local inputs and the modes involved in the swap operation.

Terminology

Entanglement — 1× in this paper · explained in 4 episodes

Summary

The gist

Entanglement between two local multi-photon multi-mode systems can be generated by swapping a pair of modes between the two local multi-mode systems.

Introduction to Mode Swapping and Entanglement Generation

Entanglement is an essential quantum resource that characterizes non-classical correlations of light, and multi-photon interference plays a crucial role in its generation. The large-scale implementation of quantum information technologies, including quantum computing, quantum communication, and quantum metrology requires increasingly large and complex linear-optical circuits capable of manipulating many optical modes and photons. Multi-photon interference is therefore best understood as a consequence of the input output mode transformation. The mode-swapping operation extends the HOM effect to spatially separated multi-mode systems. In the mode-swapping implementation introduced in previous work, one spatial mode from each local system is exchanged between the two systems. As a result, non-local two photon interference and photon number correlations arise between the two spatially separated systems.

Optical Setup and Mode-Swapping Transformation

The interferometer consists of two systems, A and B, each containing three local modes. A local unitary transformation is applied to each system before the mode-swapping operation, and the inverse transformation is applied after the swap. When one pair of modes is exchanged in the intermediate paths, however, non-local correlations are introduced between the two systems. The creation operators of the three modes are denoted by aˆ†m for systems A and bˆ†m for systems B. The local unitary transformations are chosen such that the modes sˆA and ˆsB exchanged in the intermediate paths correspond to unbiased superpositions of the input modes of the respective local systems. This relation can be realized by applying the following discrete Fourier transform (DFT). The DFT ensures that the modes sˆA and sˆB exchanged in the intermediate paths are equal superpositions of the input modes of each local system.

Analysis of Post-Selected Three-Photon State

The analysis focuses on an asymmetric three-photon input state Ψin⟩ = 200⟩A 100⟩B = 1/sqrt(2) aˆ†02 bˆ†0vac⟩. The final output state is obtained by applying the swap unitary operation Uˆswap to the input state. The resulting state after passing through the interferometer is ΨMS⟩ = 1/sqrt(2) aˆ†0 + 1/sqrt(3) sˆ†B - sˆ†A2 bˆ†0 - 1/sqrt(3) sˆ†B - sˆ†Avac⟩. The initial local photon number distribution (NA, NB) = (2, 1) is obtained when the changes in the photon numbers contained in the swapped modes of the two systems cancel each other. After post-selection, only the term without photon transfer and the term describing the reciprocal transfer of one photon in each direction remain. The success probability of this post-selection is P2,1 = 8/27 + 4/27 = 4/9.

Conclusions on Entanglement Generation

The mode swapping operation is restricted to the two-mode subspace span by the two swapped modes, and its effect appears through the difference between the two swapped-mode operators, ˆsˆ†B - sˆ†A. The post-selected state Ψ2,1⟩ is obtained after normalizing and expanding it in the Fock basis of the external modes. The joint photon number probabilities P(nA, nB) for the input state 200⟩A 100⟩B are shown in Table 1. The conditional probability that the output photon number distribution is the same as the input distribution 200⟩A 100⟩B is 4/9, where only results with NA = 2 and NB = 1 are considered. The zero probabilities of 200⟩A 010B and 200A 01B indicate that complete destructive interference occurs between the no-swap and swap amplitudes. This indicates the potential of entanglement generation by mode swapping between two multi-mode multi-photon systems. The theoretical framework developed here provides the basis for extending the analysis to optical quantum circuits with larger numbers of modes and photons.

Acknowledgments and References

This work was supported by ERATO, Japan Science and Technology Agency (JPMJER2402), and by JST SPRING, Grant Number JPMJSP2132. The references listed provide foundational context for the study of multiphoton entanglement and linear optical quantum computing.

--- Page 1 ---

Scaling up multi-mode entanglement generated by mode swapping Yuki Kodama a and Holger F. Hofmann a Graduate School of Advanced Science and Engineering, Hiroshima University, Higashi-Hiroshima 739-8527, Japan ABSTRACT Entanglement between two local multi-photon multi-mode systems can be generated by swapping a pair of modes between the two local multi-mode systems. In this presentation, we consider possible strategies for the efficient generation of entanglement between systems with three or more modes. It is shown that the choice of photon number inputs in the local systems adds a new degree of freedom to the non-local interference effects observed in the output photon number statistics. Keywords: mode swapping, multi-photon interference, multi-mode entanglement, non-local optical modes, discrete Fourier transform, linear optics.

--- Page 2 ---

absence of mode swapping, the two local transformation cancel each other. Therefore, any change in the output relative to the input state can be directly attributed to the mode swapping operation. In Section 2, we define the swapped modes as linear combinations of the local modes and show how mode swapping connects each local output mode to modes in the other system. In Section 3, we calculate the output state and the photon number distribution for the specific asymmetric three photon input state 200⟩A 100⟩B and discuss the multiphoton interference and the correlations between two system generated by mode swapping.

--- Page 3 ---

The creation operators of the three modes are denoted by aˆ†m for systems A and bˆ†m for systems B, respectively. In the present setup, each local system consists of three modes, and therefore M = 3. As mentioned above, the DFT ensures that the modes sˆA and sˆB exchanged in the intermediate paths are equal superpositions of the input modes of each local system. If the swapped modes are the k = 0 outputs of the DFT, they can be expressed as sˆ†A = 1/sqrt(M) M(X-1)/m=0 aˆ†m, sˆ†B = 1/sqrt(M) M(X-1)/m=0 bˆ†m. The first term is orthogonal to ˆsA.

Improvements for AI systems

Confidence in quantum state manipulation: The derived transformation, Uˆswapaˆ†mUˆ †swap = ˆa†m + 1/√M sˆ†B − sˆ†A, demonstrates that the action of mode swapping is restricted to the two-mode subspace span by the two swapped modes. This allows for designing quantum circuits where entanglement generation is explicitly mapped onto a controlled, localized operation between specific output modes.
Quantum correlation measurement: The analysis shows that "the local photon number distribution (2, 1) is preserved either when all three photons are contained in components orthogonal to the swapped modes, so that no photon transfer occurs between the systems, or when one photon is exchanged in each direction between systems A and B." An improved AI system could use this insight to optimize post-selection protocols for quantum communication tasks where maintaining specific local photon number correlations is critical.
Entanglement verification: The paper demonstrates the generation of entanglement using an asymmetric three-photon input state 200⟩A 100⟩B, and the resulting state exhibits strong correlations in the output photon number distributions between the input modes of the two local systems. This provides a blueprint for developing AI algorithms capable of detecting subtle, non-classical correlations in complex multi-mode photonic outputs, which is crucial for quantum metrology applications.

Sources

The non-local Hong-Ou-Mandel effect